N OU < V4 R0 2 
ak 
- D 
pda 1 
" 
€ (OQ "^ = U4 0 LR = 
In general, MAP estimator is computationally efficient and is 
not sensitive to irregular sampling. However, it obeys the 
Rayleigh resolution limit, i.e. it has almost no super-resolution 
capability. 
32.2 Nonlinear Least Squares (NLS) 
Assuming the presence of K scatterers inside a pixel with 
elevations of s=[s, 8a, ae the under-determined 
system model (1) reduces to the following over-determined 
problem: 
$us =(R*(s)R(s)) R"(s)g 3) 
For Gaussian white noise, NLS is identical to the maximum 
likelihood estimator (MLE). It is therefore theoretically the best 
estimator for our application if and only if the data closely 
agree with the assumed model. Due to the large computational 
effort to the multi-dimensional search, the NP-hard NLS is not 
recommended for practical data processing. 
3.2.3 SLIMMER 
The "Scale-down by L1 norm Minimization, Model selection, 
and Estimation Reconstruction" (SLIMMER, pronounced 
"slimmer") algorithm is a spectral estimator firstly proposed in 
(Zhu and Bamler, 2010b). It consists of three main steps: 1) a 
dimensionality scale-down by L1 norm minimization, 2) model 
selection and 3) linear parameter estimation. In case there is no 
prior knowledge about the number of scatterers inside the pixel 
and in the presence of measurement noise, the sparse 
reconstruction of (1) is give by the following L1- L2 norm 
minimization: 
  
  
adr) @ 
The L;-L, norm minimization step shrinks R dramatically and 
gives a first sparse estimate of y. Due to the fact the following 
two effects (Zhu, 2011): 1) for our application, RIP and 
incoherence are violated for several reasons 2) - The L1 norm 
approximation of the NP-hard LO norm regularization 
introduces amplitude bias, This estimate may still contain the 
outliers. Therefore, a further model order selection and 
parameter estimation step are followed to refine the sparse 
estimates obtained from (5). 
SLIMMER offers an aesthetic non-parametric realization of the 
NP-hard NLS estimator. As an efficient estimator, it is 
demonstrated to provide a super-resolution capability reaches 
the fundamental bounds of all spectral estimators. 
Younes = 918 min {lg -Ry 
3.3 Tomographic SAR Reconstruction with Colored Noise 
Based on the discussion in Section 2.2, i.e. the different data 
quality possessed by repeat- and single-pass data can be 
therefore characterized by the noise variance matrix C,, , the 
aforementioned estimators are extended to the colored noise 
case as following: 
e MAP Estimator 
f,» -(R" C2 R41) R*Czg (5) 
* Nonlinear Least Squares (NLS) 
faus-(R'(sCZR(s) R*()C2g © 
e SLIMMER 
hu cargmin((g-Ry) C?(g-Rr)e2.|rh] — C 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B7, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
3.4 Cramér-Rao Lower Bound of the Elevation Estimates 
Under colored noise, the Cramér-Rao Lower Bound (CRLB) of 
the elevation estimates in presence of only a single scatterer is 
given by: 
Ar 
S mr E (8) 
4r 42 |V SNR, 6, 
where SNR, stands for the signal-to-noise ratio (SNR) of the 
n? acquisition (n 21,...,N ). 6, is the re-weighted standard 
deviation of the baseline distribution: 
Y.SNR,b; (Y,SNR,b 
Ge | Az — (9) 
YSNR, | ESNR, 
Compared to the CRLB under Gaussian white noise 0 (i.e. all 
data possesses the same SNR ), the N- SNR dependent factor is 
replaced by D SNR, to account for variant data quality and the 
standard deviation of the baseline distribution o, is 
accordingly replaced by the &, depends on the sampling 
position b, re-weighted according to the data quality SNR, . It 
tells that, in case of adding several high quality TanDEM-X 
data to an existing TerraSAR-X stack, the elevation estimation 
accuracy improvement depends not only on the number of 
TanDEM-X images, but also the corresponding baseline 
distribution. Le. widely spread baselines are preferred. 
Considering a mixed stack consisting of N, images with a 
higher SNR of SNR, and N, images with a lower SNR of 
SNR, , let’s assume the standard deviation of the baseline 
distribution for each sub-stack are o,, and o,, , eq. (8) can be 
approximated by: 
o EE (10) 
YW ve, 
where, 
i=12 (11) 
Ar 
GET) 
"4n A2 JN, SNR, o,, 
4. EXPERIEMENNTAL RESULTS 
A reasonable data quality assumption for a mixed repeat- and 
single-pass data stack is that we have a mixed stack consisting 
of N, images with a higher SNR of SNR, and N, images 
with a lower SNR of SNR, . 
The data is simulated using the regularly distributed elevation 
aperture shown in Fig.5 (25 images, elevation resolution 
p, = 40.5m ) with the following two cases: 
e TSX: multi-pass data stack case, i.e. 25 images with 
SNR=5dB; 
e TDX: mixed stack, 20 images with SNR of 5dB 
(elevation aperture positions marked as black) and 5 
images with SNR=20dB (marked as green). 
Fig.6 shows comparison of the reflectivity profiles along 
elevation direction reconstructed by MAP (blue) and 
SLIMMER (red) using the data stack of TSX and TDX cases 
(right). The true elevations of the two scatterers are 10 and 25 
meter, respectively, an i.e. elevation distance is 0.4 times of the 
Rayleigh resolution limit. It is obvious to see the better sidelobe 
suppression for the non-parametric estimator and better super-